This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...

The nominal approach to abstract syntax deals with the issues of bound names and α-equivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo α-equivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higher-order logic of principles of α-structural recursion and induction for α-equivalence classes from the ordinary versions of these principles for abstract syntax trees.

Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.

We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , name-binding, and unique choice of fresh names. The Schanuel topos --- the category underlying FM set theory --- is an instance of this axiomatisation.

Abstract. There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and name-binding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax trees modulo α-equivalence. At the heart of this formalisation is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higher-order logic of principles of α-structural recursion and induction for α-equivalence classes from the ordinary versions of these principles for abstract syntax trees. 1

Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques